Question

The electron of hydrogen atom is considered to be revolving round a proton in circular orbit of radius \(h^2/me^2\) with velocity \(e^2/h\), where \(h = h/2\pi\). The current \(i\) is

• A. \( \dfrac{4\pi^2 m e^5}{h^2} \)
• B. \( \dfrac{4\pi^2 m e^5}{h^3} \)
• C. \( \dfrac{4\pi^2 m^2 e^2}{h^3} \)
• D. \( \dfrac{4\pi^2 m^2 e^5}{h^3} \)

Hint:

Current due to revolving charge: \(i=\dfrac{ev}{2\pi r}\). Substitute \(r\) and \(v\) and simplify carefully.

Answer 1

The Correct Option is B

Step 1: Current due to revolving electron.
An electron moving in circular orbit forms a current:
\[ i = \frac{e}{T} \] where \(T\) is time period.
Step 2: Time period of revolution.
\[ T = \frac{2\pi r}{v} \] So,
\[ i = \frac{e}{2\pi r/v} = \frac{ev}{2\pi r} \] Step 3: Substitute given values of \(r\) and \(v\).
\[ r = \frac{h^2}{me^2},\quad v = \frac{e^2}{h} \] \[ i = \frac{e\left(\frac{e^2}{h}\right)}{2\pi\left(\frac{h^2}{me^2}\right)} \] Step 4: Simplify.
\[ i = \frac{e^3}{h} \cdot \frac{me^2}{2\pi h^2} = \frac{m e^5}{2\pi h^3} \] Now converting into \(\hbar\) form gives:
\[ i = \frac{4\pi^2 m e^5}{h^3} \] which matches option (B).
Final Answer: \[ \boxed{\dfrac{4\pi^2 m e^5}{h^3}} \]